Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Transcription

1 Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions

2 Definition Walrasian Demand Given a utility function u : R n + R, the Walrasian demand correspondence x : R n ++ R + R n + is defined by x (p, w) = arg max x B p,w u(x) where B(p, w) = {x R n + : p x w}. Exercise Assumptions: goods are perfectly divisible; consumption is non negative; income is non negative; prices are strictly positive; the total price of consumption cannot exceed income; prices are linear in consumption. Think of possible violations. Suppose w = $100. There are two commodities, electricity and food. Each unit of food costs $1. The first 20Kwh electricity cost $1 per Kwh, but the price of each incremetal unit of electricity is $1.50 per Kwh. Write the consumer s budget set formally and draw it.

3 Proposition Walrasian Demand Is Convex If u is quasiconcave, then x (p, w) is convex. If u is strictly quasiconcave, then x (p, w) is unique. Proof. Same as before (u (stricly) quasiconcave means (strictly) convex). Suppose x, y x (p, w) and pick α [0, 1]. First convexity: need to show αx + (1 α)y x (p, w). x y by definition of x (p, w ). u is quasiconcave, thus is convex and αx + (1 α)y y. y z for any z B(p, w ) by definition of x (p, w ). Transitivity implies αx + (1 α)y z for any z B(p, w ); thus αx + (1 α)y x (p, w ). Now uniqueness. x, y x (p, w ) and x y imply αx + (1 α)y y for any α (0, 1) because u is strictly quasiconcave ( is strictly convex). Since B(p, w ) is convex, αx + (1 α)y B(p, w ), contradicting y x (p, w ).

4 Walrasian Demand Is Non-Empty and Compact Proposition If u is continuous, then x (p, w) is nonempty and compact. We already proved this as well. Proof. Define A by A = B(p, w) = {x R n + : p x w} This is a closed and bounded (i.e. compact, set) and x (p, w) = C (A) = C (B(p, w)) where are the preferences represented by u. Then x (p, w) is the set of maximizers of a continuous function over a compact set.

5 Walrasian Demand: Examples How do we find the Walrasian Demand? Need to solve a constrained maximization problem, usually using calculus. Question 2, Problem Set 3; due next Monday. For each of the following utility functions, find the Walrasian demand correspondence. (Hint: pictures may help) 1 u(x) = n i=1 x αi i with α i > 0 (Cobb-Douglas). 2 u(x) = min{α 1 x 1, α 2 x 2,..., α n x n } with α i > 0 (generalized Leontief). 3 u(x) = n i=1 α i x i for α i > 0 (generalized linear). 4 u(x) = [ n i=1 α i x ρ i ] 1 ρ (generalized CES). Can we do the second one using calculus? How about the third? Do we need calculus? Always draw a picture. Constant elasticity of substitution (CES) preferences are the most commonly used homothetic preferences. Many preferences are a special case of CES.

6 An Optimization Recipe How to solve max f (x) subject to g i (x) 0 with i = 1,.., m 1 Write the Langrange function L : R n R m R as m L (x, λ) = f (x) λ i g i (x) 2 Write the First Order Conditions: n 1 { }} { L (x, λ) = f (x) i=1 m λ i g i (x) = 0 i=1 } {{ } f (x) x j m i=1 g λ i (x) i x =0 for all j=1,..,n j 3 Write constraints, inequalities for λ, and complementary slackness conditions: g i (x) 0 with i = 1,.., m λ i 0 with i = 1,.., m λ i g i (x) = 0 with i = 1,.., m 4 Find the x and λ that satisfy all these and you are done...hopefully.

7 The Recipe In Action: Cobb-Dougals Utility Compute Walrasian demand when the utility function is u(x 1, x 2 ) = x α 1 x 1 α 2 Here x (p, w) is the solution to 1 The Langrangian is max x 1 α x 1 α 2 x 1,x 2 {p 1x 1+p 2x 2 w, x 1 0, x 2 0} L (x, λ) = x1 α x 1 α 2 λ w (p 1 x 1 + p 2 x 2 w) ( λ 1 x 1 ) ( λ 2 x 2 ) 2 The First Order Conditions for x is: L (x, λ) = {{ } 2 1 αx α 1 1 x 1 α 2 λ w p 1 + λ 1 (1 α) x α 1 x α 2 λ w p 2 + λ 2 = α u(x1,x2) x 1 λ w p 1 + λ 1 (1 α) u(x1,x2) x 2 λ w p 2 + λ 2 3 The constraints, inequalities for λ, and complementary slackness are: p 1 x 1 + p 2 x 2 w 0 x 1 0, and x 2 0 λ w 0, λ 1 0, and λ 2 0 λ w (p 1 x 1 + p 2 x 2 w) = 0, λ 1 x 1 = 0, and λ 2 x 2 = 0 4 Find a solution to the above (easy for me to say). = 0

8 The Recipe In Action: Cobb-Dougals Utility Compute Walrasian demand when the utility function is u(x 1, x 2 ) = x1 αx 1 α 2 We must solve: α u(x1,x2) x 1 λ w p 1 + λ 1 = 0 and (1 α) u(x1,x2) x 2 λ w p 2 + λ 2 = 0 p 1 x 1 + p 2 x 2 w 0 x 1 0, x 2 0 and λ w 0, λ 1 0, λ 2 0 λ w (p 1 x 1 + p 2 x 2 w) = 0 and λ 1 x 1 = 0, λ 2 x 2 = 0 x (p, w) must be strictly positive (why?), hence λ 1 = λ 2 = 0. The budget constraint must bind (why?), hence λ w 0. Therefore the top two equalities become αu(x 1, x 2 ) = λ w p 1 x 1 and (1 α) u(x 1, x 2 ) = λ w p 2 x 2 Dividing one by the other and use Full Expenditure. α 1 α = p1x1 p 2x 2 Summing both sides and using Full Expenditure we get Some algebra then yields u(x 1, x 2 ) = λ w (p 1 x 1 + p 2 x 2 ) = λ w w x1 (p, w) = αw, x (1 α) w 2 (p, w) =, and λ w = p 1 p 2 ( ) α ( α 1 α p 1 p 2 ) 1 α

9 Marginal Rate of Substitution The First Order Conditions for utility maximization take the form: utility ( function ) λ budget budget ( constraint constraint ) non negativity λ non negativity ( constraints constraints ) = 0 If the non-negativity constraints hold, the correspoding multipliers equal 0. Then, at a solution x, the expression above is: u (x ) x j λ w p j = 0 for all j = 1,.., n One way to rearrange this expression gives u(x ) x j u(x ) x k = p j p k for any j, k This is the familiar condition about equalizing marginal rates of substituitons across goods.

10 Marginal Utility Per Dollar Spent Again, the First Order Conditions for utility maximization take the form u (x ) x j λ w p j λ j = 0 for all j = 1,.., n Another way to rearrange it gives u(x ) x j = λ w + λ j If not, there are goods j and k for which for all j = 1,.., n p j p j In words, if any two goods are consumed in strictly positive amounts, the marginal utility per dollar spent must be equal across them. u(x ) u(x ) x j p j < x k p k Then DM can spend ε less on j and ε more on k so that the budget constraint still holds (why?). By Taylor s theorem, the utility at this new choice is u (x )+ u (x ) x j ( εpj ) + u (x ) x k ( ) ε +o (ε) = u (x )+ε p k u(x ) x k p k which implies that x is not an optimum. Think about the case in which some goods are consumed in zero amount. u(x ) x j + p j

11 Luca s Rough Guide to Convex Optimization Where does the recipe come from? Roee told you, but here is my informal summary. Question Let f : R n + R be a continuous, increasing, and quasi-concave function, and let C R n be a convex set. We want to find a solution to the following problem: where max f (x) x C C = {x R n : g i (x) 0 with i = 1,..., N} This is the most general way to state a maximization problem. Examples 1 If C = R n, we have an unconstrained problem. 2 If you have constraints like h (x) b, define g j (x) = h (x) b. 3 If you have constraints like h (x) b, define g j (x) = [h (x) b]. 4 If you have constraints like h (x) = b, define g j (x) = h (x) b and g k (x) = [h (x) b] and rewrite as {g j (x) 0 and g k (x) 0}.

12 Simple Geometry Observations A level curve for some function f : R n + R is given by f (x) = c for some c R. The better than set is (x) = {y R n : f (y) f (x)}. Draw C and some level curves (when are better than sets convex?). The tangent plane to C at a point is a plane through the point that does not intersect the interior of C. The tangent to the level curves at a point is a plane through the point that does not intersect the interior of (x). At a maximum, level curves and constraint set are tangent to the same plane.

13 Geometry In Action Definition An hyperplane is H = {x R n : γ (x y) = 0}. Fix x on the boundary of C, the tangent to C at x is a supporting hyperplane. Definition An hyperplane H supports C at a point x if C is a strict subset of H = {x R n : γ (x y) 0} The tangent to the better than set at x is also a supporting hyperplane. Definition An hyperplane H supports the better than set at x if (x) is a strict subset of H + = {x R n : γ (x y) 0} Supporting Hyperplane Theorem A non-empty convex set has at least one supporting hyperplane. An optimum is a point x where the same hyperlane supports C and (x).

14 Details To Remember If the better than set or the constraint sets are not convex: big trouble. If functions are not differentiable: small trouble. If the geometry still works we can find a more general theorem (see convex analysis). When does the recipe fail? If the constraint qualification condition fails. If the objective function is not quasi concave. This means you must check the second order conditions when in doubt. these have to do with the matrix of second derivatives; look at Section M.K in MWG for details (boring and mechanical, but one needs to know them); Varian also has a good quick and dirty guide.

15 Summary Convex Optimization Summary When x is a solution of the problem max f (x) subject to g i (x) 0 i = 1,.., m then with m L (x, λ) = f (x ) λ i g i (x ) = 0 i=1 λ i 0 and λ i g i (x ) = 0 for each i = 1,.., m and m Dx 2 L (x, λ) = D 2 f (x ) λ i D 2 g i (x ) is negative semidefinite on {z R n : g i (x ) z = 0 for each i = 1,.., m}......provided constraint qualification holds. i=1

16 Demand and Indirect Utility Function The Walrasian demand correspondence x : R n ++ R + R n + is defined by x (p, w) = arg max u(x) where B(p, w) = {x x B(p,w ) Rn + : p x w}. Definition Given a continuous utility function u : R n + R, the indirect utility function v : R n ++ R + R is defined by Results v(p, w) = u(x (p, w)) where x (p, w) arg max x B(p,w ) u(x). This indirect utility function measures changes in the optimized value of the objective function as the parameters (prices and wages) change and the consumer adjusts her optimal consumption accordingly. The Walrasian demand correspondence is upper hemi continuous To prove this we need properties that characterize continuity for correspondences. The indirect utility function is continuous. To prove this we need properties that characterize continuity for correspondences.

17 Berge s Theorem of the Maximum The theorem of the maximum lets us establish the previous two results. Theorem (Theorem of the Maximum) If f : X R is a continuous function and ϕ : Q X is a continuous correspondence with nonempty and compact values, then the mapping x : Q X defined by x (q) = arg max x ϕ(q) f (x) is an upper hemicontinuous correspondence and the mapping v : Q R defined by is a continuous function. v(q) = max x ϕ(q) f (x) Berge s Theorem is useful when exogenous parameters enter the optimization problem only through the constraints, and do not directly enter the objective function. This happens in the consumer s problem: prices and income do not enter the utility function, they only affect the budget set.

18 Properties of Walrasian Demand Proposition If u(x) is continuous, then x (p, w) is upper hemicontinuous and v(p, w) is continuous. Proof. Apply Berge s Theorem: If u : R n + R a continuous function and B (p, w) : R n ++ R + R n + is a continuous correspondence with nonempty and compact values. Then: (i): x : R n ++ R + R n + defined by x (p, w) = arg max x B(p,w ) u(x) is an upper hemicontinuous correspondence and (ii): v : R n ++ R + R defined by v(p, w) = max x B(p,w ) u(x) is a continuous function. We need continuity of the correspondence from price-wage pairs to budget sets. We must show that B : R n ++ R + R n + defined by B(p, w ) = {x R n + : p x w } is continuous and we are done.

19 Continuity for Correspondences Reminder from math camp. Definition A correspondence ϕ : X Y is upper hemicontinuous at x X if for any neighborhood V Y containing ϕ(x), there exists a neighborhood U X of x such that ϕ(x ) V for all x U. lower hemicontinuous at x X if for any neighborhood V Y such that ϕ(x) V, there exists a neighborhood U X of x such that ϕ(x ) V for all x U. A correspondence is upper (lower) hemicontinuous if it is upper (lower) hemicontinuous for all x X. A correspondence is continuous if it is both upper and lower hemicontinuous.

20 Conditions for Continuity (from Math Camp) The following suffi cient conditions are sometimes easier to use. Proposition (A) Suppose X R m and Y R n. A compact-valued correspondence ϕ : X Y is upper hemicontinuous if, and only if, for any domain sequence x j x and corresponding range sequence y j such that y j ϕ(x j ), there exists a convergent subsequence {y jk } such that lim y jk ϕ(x). Note that compactness of the image is required. Proposition (B) Suppose A R m, B R n, and f : A B. Then ϕ is lower hemicontinuous if, and only if, for all {x m } A such that x m x A and y ϕ(x), there exist y m ϕ(x m ) such that y m y.

21 Continuity for Correspondences: Examples Exercise Suppose ϕ : R R is defined by: ϕ(x) = { {1} if x < 1 [0, 2] if x 1. Prove that ϕ is upper hemicontinuous, but not lower hemicontinuous. Exercise Suppose ϕ : R R is defined by: ϕ(x) = { {1} if x 1 [0, 2] if x > 1. Prove that ϕ is lower hemicontinuous, but not upper hemicontinuous.

22 Continuity of the Budget Set Correspondence Question 4, Problem Set 3; due next Monday. Show that the correspondence from price-wage pairs to budget sets, B : R n ++ R + R n + defined by is continuous. B(p, w) = {x R n + : p x w} First show that B(p, w) is upper hemi continuous. Then show that B(p, w) is lower hemi continuous. In both cases, use the propositions in the previous slide (and Bolzano-Weierstrass). There was a very very similar problem in math camp (PS 6, #9).

23 Econ 2100 Fall 2015 Problem Set 3 Due 21 September, Monday, at the beginning of class 1. Prove that if u represents, then: (a) is weakly monotone if and only if u is nondecreasing; is strictly monotone if and only if u is strictly increasing. (b) is convex if and only if u is quasiconcave; is strictly convex if and only if u is strictly quasiconcave. (c) u is quasi-linear if and only 1. (x, m) (x, m ) if and only if m m, for all x R n 1 and all m, m R; 2. (x, m) (x, m ) if and only if (x, m+m ) (x, m +m ), for all x R n 1 and m, m, m R; 3. for all x, x R n 1, there exist m, m R such that (x, m) (x, m ). (d) a continuous is homothetic if and only if it is represented by a utility function that is homogeneous of degree Consider a setting with 2 goods. For each of the following utility functions, find the Walrasian demand correspondence. (Hint: draw pictures) (a) u(x) = n i=1 x α i i with α i > 0. (Cobb-Douglas) (b) u(x) = min{α 1 x 1, α 2 x 2,..., α n x n } with α i > 0. (generalized Leontief) (c) u(x) = n i=1 α ix i with α i > 0. (linear) ] 1 ρ (d) u(x) = [ n i=1 α ix ρ (CES) i From now on consider CES preferences when n = Show that CES preferences are homothetic. 2. Show that these preferences become linear when ρ = 1, and Leontieff as ρ. 3. Assume strictly positive consumption and show that these preferences become Cobb-Douglas as ρ 1. [Hint: use L Hopital Rule] 4. Compute the Walrasian Demand when ρ < Verify that when ρ < 1 the Walrasian Demand converges to the Walrasian Demands for Leontieff and Cobb-Douglas utility functions as ρ and ρ 0 repectively. 6. The elasticity of substitution between x 1 and x 2 is ξ 1,2 = [ ] x 1 (p,w) x 2 (p,w) p 1 p 2 p 1 p 2 x 1 (p,w) x 2 (p,w) Prove that for CES preferences ξ 1,2 = 1 1 ρ. What is ξ 1,2 for linear, Leontieff, and Cobb- Douglas preferences? 1

24 3. (Properties of Differentiable Walrasian Demand) Assume that the Walrasian demand x (p, w) is a differentiable function. (a) Show that for each (p, w), n i=1 x k (p, w) p i + w x k (p, w) p i w = 0 for k = 1,..., n Hint: x is homogeneous of degree 0 in (p, w). What does this imply? (This is a special case of Euler s formula for functions that are homogeneous of degree r. See MWG M.B for more.) (b) Suppose in addition that is locally nonsatiated. Show that for each (p, w), n k=1 x k (p, w) p k + x i (p, w) = 0 for i = 1,..., n and p i n k=1 x k (p, w) p k w = 1. Give a simple intuitive (short) version of these two results (sometimes called Cournot and Engel aggregation, respectively) in words. 4. Prove that the correspondence from price-wage pairs to budget sets, B : R n ++ R + R n + defined by is continuous. B(p, w) = {x R n + : p x w} 5. Suppose a consumer can choose to move to either Philadelphia or Queens. The same n goods are available in both locations, and he must make all of his purchases in the location where he chooses to move. The prices may differ, however; let p denote the price vector in Philadelphia, and let q denote the price vector in Queens. Notice that he can decide where to move, but he cannot purchase some goods in Philadelphia and other goods in Queens. (a) Describe the consumer s budget set B(p, q, w) for fixed p, q R n ++ and wealth w R +. (b) Suppose the consumer s utility function u : R n + R is continuous. Prove the following: The Walrasian demand correspondence, which is now defined on two prices and wealth, x : R n ++ R n ++ R + R n +, defined by x (p, q, w) = arg max u(x) x B(p,q,w) is upper hemicontinuous; and the indirect utility function, v : R n ++ R n ++ R + R, defined by v(p, q, w) = max u(x) x B(p,q,w) is continuous. (c) Prove or disprove the following: If the consumer s utility function is strictly quasiconcave, then x (p, q, w) 1. 2